On the topological aspects of the theory of represented spaces∗

Arno Pauly Clare College University of Cambridge, United Kingdom [email protected]

Represented spaces form the general setting for the study of computability derived from Turing ma- chines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by ESCARDO´ ,SCHRODER¨ , and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and – closely linked to these – properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.

1 Introduction

∗ Just as numberings provide a tool to transfer computability from N (or {0,1} ) to all sorts of countable structures; representations provide a means to introduce computability on structures of the cardinality of the continuum based on computability on Cantor space {0,1}N. This is of course essential for com- n m k n m putable analysis [66], dealing with spaces such as R, C (R ,R ), C (R ,R ) or general Hilbert spaces [8]. Computable measure theory (e.g. [65, 58, 44, 30, 21]), or computability on the set of countable ordinals [39, 48] likewise rely on representations as foundation. Essentially, by equipping a set with a representation, we arrive at a represented space – to some extent1 the most general structure carrying a notion of computability that is derived from Turing ma- chines2. Any represented space provides in a canonic way also function spaces and spaces of certain subsets. While the usefulness of these derived spaces will vary depending on the area of application, their development can be done while abstracting away from the specifics of the original represented space – in fact, such details have traditionally obfuscated the core proof ideas. Closely linked to the var- ious subset spaces, also a few properties of spaces defined in terms of computability of certain maps are studied. Among these, special attention shall be drawn to admissibility, which has often been advocated as a criterion for well-behaved representations in computable analysis [66, 56]. arXiv:1204.3763v3 [math.LO] 2 Mar 2015 ∗Prior versions of this manuscript were titled Compactness and Separation for Represented Spaces and A new introduction to the theory of represented spaces. 1The notion of a multi-representation [55, 68] goes beyond representations. A multi-representation of a set X is a right-total relation δ ⊆ {0,1}N × X, relating codes to the encoded elements. The distinction to representation is that a code can stand for more than one element here. Several common examples of multi-representations have the additional feature that ∃p ∈ {0,1}Nδ(p,x) ∧ δ(p,y) defines an equivalence relation on X – hence, we can conceive of the multi-representation as an ordinary representation of the induced equivalence classes. Often we even have a canonization operation available; such as the saturation used in Section 5 and the closure used in Section 7 to obtain represented spaces. A very different example (communicated to the author by BAUER) is the multirepresentation ∈ : {0,1}N ⇒
P({0,1}N) \{/0} . Examples of this kind, however, are beyond the scope of the paper. 2Algebraic computation models [6] as abstract generalizations of register machines are a completely different thing. 2 Theory of Represented Spaces

2 Connections to the literature

It is well-known that many representations in computable analysis can be characterized by extremal properties – the prototypic example being that the standard representation of the space of continuous functions carries exactly as much information as needed to make function evaluation computable. In fact, this condition not only characterizes the representation, but also the set itself: Whenever a function is contained in a function space admitting computable function evaluation, it is computable w.r.t. some oracle – its name – hence, it is continuous. Consequently, rather than speaking merely about representa- tions (of fixed sets) being characterized such, we should consider represented spaces – the combination of a set and a representation of it – as the fundamental objects of the characterization results3. In many characterization results in the literature the representation is defined explicitly first, then the extremal property is proven. However, as a consequence of the UTM theorem, this is unnecessary: The condition itself defines the represented space up to computable isomorphisms. In fact, specifying details of standard representations beyond their characterizing property often complicates proofs of basic results, and can obfuscate the algorithmic ideas. Developing the basic theory of represented spaces based on extremal characterizations amounts to an instantiation of ESCARDO´ ’s synthetic topology [25, 28] with the category of represented spaces. As this category is particularly well-behaved, we obtain a very concise picture, and can prove a few more results on compactness and separation than available in the generic setting. Similarly, there are many parallels to TAYLOR’s abstract stone duality [60, 61], in comparison we again sacrifice generality for simplicity of presentation and strength of some results. Most definitions and many results about them were already formulated by SCHRODER¨ [54] before the advent of synthetic topology though. In this development, spaces were always required to satisfy some form of admissibility4. For our purposes, these restrictions are mostly immaterial, and will be dropped. Note that admissibility (w.r.t. a topology) turns out to be a feature of synthetic topology in general, demarking those spaces actually fully understandable in terms of their (internal) topology. For represented spaces, this can be seen as (slightly) generalizing [56]. The PhD thesis of LIETZ [40] also contains relevant results on admissibility, in particular [40, Theorem 3.2.7.]. A direct application of the framework of synthethic topology to represented spaces was performed by COLLINS [19, 20] in 2010, several of the result presented here are already present in [20]. Effective compactness has been studied in [16] by BRATTKA and WEIHRAUCH, and in [15] by BRATTKA and PRESSER. [71] by GRUBBA and WEIHRAUCH considers representations of closed, open and compact sets characterized by extremal properties for the restricted case of countably based spaces. Effective topological separation was considered in [69, 70] by WEIHRAUCH. A warning is due regarding a discrepancy in nomenclature: The computable elements of the space of closed subsets introduced here (i.e. the computable closed sets) are called co-c.e. closed sets by WEIHRAUCH and others. Their c.e. closed sets are referred to as overt here (a term coined by TAYLOR, see Section 7), as they as a space lack the structure expected from closed sets. Subsequently, the sets called computably closed in WEIHRAUCH’s terminology are called computable closed and overt here. The observation that the c.e. closed sets lack the closure properties of closed sets spawned various investigations into more restricted setting remedying the issue. An example of this is ZIEGLER’s work on representations for regular closed subsets of a Euclidean space [73]. A second example, somewhat

3Hence the chance of the term of choice from KREITZ’ and WEIHRAUCH’s Theory of Representations [37] to the present paper’s Theory of Represented Spaces. 4The form of admissibility presented in this paper corresponds to admissibility w.r.t. some topology. Schroder¨ also studied admissibility w.r.t some variants of limit spaces. A. Pauly 3 further removed from representations, is the observation that computable closed sets (i.e. co-c.e. closed) satisfying some topological criterion have to be computably overt (i.e. c.e. closed) as made by MILLER and ILJAZOVIC´ [42, 41, 32, 33]. As the attribution of the definitions and results is non-trivial due to parallel independent developments using slightly different formal frameworks, the discussion of the intellectual pedigree follows separately each section. In addition to work cited there, several observations presumably are folklore. This paper is originally based on [46, Chapter 3], the PhD thesis of the author.

3 The foundations

The central notion is the represented space, which is a pair X = (X,δX ) of a set X and a partial surjection 5 δX :⊆ {0,1}N → X ( ). A multi-valued function between represented spaces is a multi-valued function between the underlying sets. For f :⊆ X ⇒ Y and F :⊆ {0,1}N → {0,1}N, we call F a realizer of f (notation F ` f ), iff δY (F(p)) ∈ f (δX (p)) for all p ∈ dom( f δX ). A map between represented spaces is called computable (continuous), iff it has a computable (continuous) realizer. Similarly, we call a point ∼ x ∈ X computable, iff there is some computable p ∈ NN with δX(p) = x. We write X = Y if the two spaces are computably isomorphic, i.e. if there is a bijection f : X → Y that is computable and has a computable inverse. For our purposes, there is no need to distinguish computably isomorphic spaces.

Warning: A priori, the notion of a continuous map between represented spaces and a continuous map between topological spaces are distinct and should not be confused!

We consider two categories of represented spaces, one equipped with the computable maps, and one equipped with the continuous maps. We call the resulting structure a category extension (cf. [45, 47]), as the former is a subcategory of the latter, and shares its structure (products, coproducts, exponentials) as we shall see next. In general, all our results have two instances, one for the computable and one for the continuous maps, with the proofs being identical. Essentially, the continuous case can be considered as the relativization of the computable one, as a function on Cantor space is continuous iff it is computable relative to some oracle. Proposition 1. For any two represented spaces X, Y there are represented spaces X × Y, X + Y and computable functions π1 : X × Y → X, π2 : X × Y → X, ι1 : X → X + Y, ι2 : Y → X + Y, such that:

1. For any pair of continuous maps f1 : Z → X, f2 : Z → Y there is a unique continuous map h f1, f2i : Z → X × Y with fi = πi ◦ h f1, f2i.

2. If f1 and f2 are computable, so is h f1, f2i.

3. For any pair of continuous maps f1 : X → Z, f2 : Y → Z there is a unique continuous map ( f1 + f2) : X + Y → Z with fi = ( f1 + f2) ◦ ιi.

4. If f1 and f2 are computable, so is f1 + f2. U The space X + Y can be obtained via (X,δX ) + (Y,δY ) = (X Y,δX+Y ) where δX+Y (0p) = δX (p) and δX+Y (1p) = δY (p). The space X × Y can be obtained via (X,δX ) × (Y,δY ) = (X × Y,δX×Y ) where δX×Y (hp,qi) = (δX (p),δY (q)).

5 Our choice of Cantor space as domain for our representations is inconsequential, often Baire space NN is used instead. A recent observation by COOK and KAWAMURA is that using the set of regular functions on natural numbers as the foundation may be useful for complexity theory [35], again, this has no impact at all on the computability theory presented here. 4 Theory of Represented Spaces

Definition 2. Given a pair of represented spaces X = (X,δX ) and Y = (Y,δY ) we define X ∧ Y := (X ∩ Y,δX ∧ δY ) where (δX ∧ δY )(hp,qi) = x iff δX (p) = x ∧ δY (q) = x. Given two represented spaces X, Y we obtain a third represented space C (X,Y) of functions from X n to Y by letting 0 1p be a [δX → δY ]-name for f , if the n-th Turing machine equipped with the oracle p computes a realizer for f . As a consequence of the UTM theorem (in the form proven by WEIHRAUCH [63]), C (−,−) is the exponential in the category of continuous maps between represented spaces, and the evaluation map is even computable.

Proposition 3. Let X = (X,δX ), Y = (Y,δY ), Z = (Z,δZ), U = (U,δU ) be represented spaces. Then the following functions are computable: 1. eval : C (X,Y) × X → Y defined by eval( f ,x) = f (x). 2. curry : C (X × Y,Z) → C (X,C (Y,Z)) defined by curry( f ) = x 7→ (y 7→ f (x,y)). 3. uncurry : C (X,C (Y,Z)) → C (X × Y,Z) defined by uncurry( f ) = (x,y) 7→ f (x)(y). 4. ◦ : C (Y,Z) × C (X,Y) → C (X,Z), the composition of functions 5. × : C (X,Y) × C (U,Z) → C (X × U,Y × Z) 6. const : Y → C (X,Y) defined by const(y) = (x 7→ y). Proof. All items follow from standard arguments on Turing machines; one merely has to verify that the Type-2 semantics are unproblematic.

1. By definition of [δX → δY ], if we apply the Turing machine with oracle specified in the [δX → δY ]- name of f to a δX -name of x, we obtain a δY -name of f (x). 2. From a Turing machine M we can compute a Turing machine M0, such that M0 on input p and oracle hq,oi simulates M on input hp,qi and oracle o. 3. and vice versa. 4. Composition of Turing machines is computable, and appropriate access to the two oracles can be ensured. As composition of realizers yields a realizer of the composition, this suffices. 5. The execution of two Turing machines in parallel can be simulated by a single one, access to the oracles can be done accordingly. Products of realizers are realizers of products. 6. This is done via a Turing machine which ignores its input and copies the oracle tape to the output tape.

Corollary 4. Let X, Y, Z be represented spaces. For any x ∈ X, the map partialx : C (X × Y,Z) → C (Y,Z) defined by partialx( f ) = (y 7→ f (x,y)) is continuous. If x is computable, then so is partialx.

Remarks

The earliest use of represented spaces as entities in their own right seems to be due to BRATTKA in 1996 [7], although the treatment there does not go beyond the introduction of computability before the setting is restricted to (countably based) topological spaces. The notion of a representation (with full generality6) precedes this by far, going back to WEIHRAUCH [63] and KREITZ and WEIHRAUCH [37]

6In a more restrictive sense, WEIHRAUCH and SCHAFER¨ used the term representation before [72]. A. Pauly 5 in 1985. The category of representation (of represented spaces) was explicitly mentioned by BAUER in 2002 [3]. That the study of computability and the study of continuity go hand in hand when it comes to the theory of representations (of represented spaces) has been noted since its beginnings in [37], and takes a very visible role e.g. in WEIHRAUCH’s books [64, 66]. That this yields two categories with the same objects and the same structure was observed by BAUER (in a slightly different setting) in 1998 [1]. The observation that continuity is relativized computability, which is fundamental for this duality, seems to be a folk result7 appearing first in the 1950’s. The operations +,×,∧ on representations are all defined already in [37], their properties as given in Proposition 1 appear e.g. in [66]. The exponential is introduced in [63] (although not named as such). The category of represented spaces is identified as cartesian closed in [3].

4 Open and closed sets

In the following, we will want to make use of two special represented spaces, N = (N,δN) and S = n N N N ({⊥,>},δS). The representation are given by δN(0 10 ) = n, δS(0 ) = ⊥ and δS(p) = > for p 6= 0 . It is straightforward to verify that the computability notion for the represented space N coincides with classical computability over the natural numbers. The Sierpinski´ space S in turn allows us to formalize semi-decidability. The computable functions f : − − N → S are exactly those where f 1({>}) is recursively enumerable (and thus f 1({⊥}) co-recursively enumerable). In general, for any represented space X we obtain two spaces of subsets of X; the space − of open sets O(X) by identifying f ∈ C (X,S) with f 1({>}), and the space of closed sets A (X) by − identifying f ∈ C (X,S) with f 1({⊥}). The properties of the spaces of open and closed sets follow from a few particular computable functions on Sierpinski´ space S and the function space properties in Proposition 3. W Proposition 5. The functions ∧,∨ : S × S → S and : C (N,S) → S are computable. Here we have ∧,∨ : S × S → S defined by ∧(>,>) = >, ∧(x,y) = ⊥ for (x,y) 6= (>,>), ∨(⊥,⊥) = W W ⊥, ∨(x,y) = > for (x,y) 6= (⊥,⊥). Moreover, is defined by ((xn)n∈N) = > iff ∃n> ∈ N s.t. xn> = >, W and ((xn)n∈N) = ⊥ otherwise. Proof. A machine realizing ∧,∨ simply writes 0s until it reads a non-zero value from one (for ∨) or W both (for ∧) input components, then it continues with 1s. In order to solve , all (xn) are investigated simultaneously, while 0s are written. If a 1 ever occurs anywhere in the input, the program starts writing 1s. V We point out that neither ¬ : S → S or : C (N,S) → S are computable. Proposition 6. Let X, Y be represented spaces. Then the following functions are well-defined and computable: 1. C : O(X) → A (X), C : A (X) → O(X) mapping a set to its complement 2. ∪ : O(X) × O(X) → O(X), ∪ : A (X) × A (X) → A (X) 3. ∩ : O(X) × O(X) → O(X), ∩ : A (X) × A (X) → A (X) S S 4. : C (N,O(X)) → O(X) mapping a sequence (Un)n∈N of open sets to their union n∈N Un 7The difficulty in attributing the result to any specific person is also mentioned in [34]. 6 Theory of Represented Spaces

T T 5. : C (N,A (X)) → A (X) mapping a sequence (An)n∈N of closed sets to their intersection n∈N An 6. −1 : C (X,Y) → C (O(Y),O(X)) mapping f to f −1 as a set-valued function for open sets 7. ∈ : X × O(X) → S defined by ∈(x,U) = >, if x ∈ U. 8. × : O(X) × O(Y) → O(X × Y), × : A (X) × A (Y) → A (X × Y) 9. Cut : Y × O(X × Y) → O(X) mapping (y,U) to {x | (x,y) ∈ U}

10. Π : C (N,A (X)) → A (C (N,X)), where Π((An)n∈N) = Πn∈NAn.

Proof. 1. By definition both functions are realized by id{0,1}N . 2. Taking into consideration that O(X), A (X) may be considered as subspaces of the function space C (X,S), we may realize ∪ : O(X)×O(X) → O(X) by composition (Proposition 3 (4, 6), together with the diagonal) of ∧ and × from Proposition 3 (5), likewise the composition of ∨ and × realizes ∪ : A (X) × A (X) → A (X). 3. This follows from 1. and 2. using de Morgan’s law. 4. Composition of W with the input yields the output, and is computable due to Proposition 3 (4). 5. This follows from 1. and 4. using de Morgan’s law. 6. Again, this is a special case of composition, which is computable due to Proposition 3 (4). 7. Here we have a special case of eval, which is computable due to Proposition 3 (1). 8. This follows from composing the computable function × : C (X,S)×C (Y,S) → C (X×Y,S×S) from Proposition 3 (6) together with ∧ : S × S → S and ∨ : S × S → S respectively. 9. This follows from Proposition 3 (1,2) via x ∈ Cut(y,U) iff (x,y) ∈ U. W 10. Essentially, this is composition with : C (N,S) → S from Proposition 5.

A represented space X canonically induces a topological space by equipping the set X by the quotient topology TX of δX :⊆ {0,1}N → X. One can verify that TX is the underlying set of O(X); i.e. that the open sets of a represented space are the open sets in the induced topological space: Essentially, the open subset of a subspace of {0,1}N that witnesses that a set U is in TX can be translated into a continuous realizer of χU : X → S. Proposition 6 (6) implies that a continuous map between represented spaces is also continuous as a map between the induced topological spaces. However, the converse is generally false.

Remarks The introduction of Sierpinski´ space to the study of representations, and subsequent use to derive rep- resentations of the open and the closed subsets, is due to SCHRODER¨ in 2002 [54] (cf. [15]). However, the usefulness of such definitions to obtain concise proofs has not been fully appreciated in this setting (e.g. WEIHRAUCH and GRUBBA are not using it in [71] from 2009). Statements equivalent to the items of Proposition 6 have been proven in various restricted settings before. Using the combination of the function space construction and the presence of Sierpinski´ space to obtain the space of open (closed) sets together with the standard operations on them is the fundamental idea of synthetic topology (ESCARDO´ 2004 [25]). A. Pauly 7

COLLINS has stated the results of Proposition 6 (1. − 6.,9.) as part of [20, Theorem 3.23, Theorem 3.24, & Proposition 3.26]. The space A (X) has been studied as the upper Fell topology on the hyperspace of closed sets in topology. This topology was introduced by FELL in 1962 [29]. Note that the characterization of compact sets in Section 5 is crucial for this coincidence, which has been observed before by BRATTKA and PRESSER 2003 [15]. A general source for hyperspace topologies is [5].

5 Compactness

Compactness is occasionally described as a generalization of finiteness, in that compactness means that it can be verified that a property applies to all elements of a space – so in same sense, compact spaces admit a form of exhaustive search8, despite potentially having uncountable cardinality. Compactness both plays a roleˆ as a property of spaces, as well as inducing a represented space of subsets of a given space. We provide various equivalent characterizations of the former, and list various useful computable operations on the latter.

Definition 7. A represented space X is (computably) compact, if the map IsEmptyX : A (X) → S defined by IsEmptyX(/0) = > and IsEmptyX(A) = ⊥ otherwise is continuous (computable). Proposition 8. The following properties are equivalent for a represented space X: 1. X is (computably) compact.

2. IsFullX : O(X) → S defined by IsFullX(X) = > and IsFullX(U) = ⊥ otherwise is continuous (com- putable). 3. For every (computable9) A ∈ A (X) the subspace A is (computably) compact. 4. ⊆: A (X) × O(X) → S defined by ⊆ (A,U) = >, iff A ⊆ U is continuous (computable). S 5. IsCover : C (N,O(X)) → S defined by IsCover((Un)n∈N) = >, iff n∈N Un = X is continuous (com- putable). 6. FiniteSubcover :⊆ C (N,O(X)) ⇒ N with dom(FiniteSubcover) S S = {(Un)n∈N | ∃N n≤N Un = X} and N ∈ FiniteSubcover((Un)n∈N) iff n≤N Un = X is continu- ous (computable). T 7. Enough :⊆ C (N,A (X)) ⇒ N with (Ai)i∈N ∈ dom(Enough) iff ∃N i≤N Ai = /0,and N ∈ Enough((Ai)i∈N) T iff i≤N Ai = /0is continuous (computable).

8. For every represented spaces Y, the map π2 : A (X × Y) → A (Y) defined by π2(A) = {y ∈ Y | ∃x ∈ X (x,y) ∈ A} is well-defined and continuous (computable).

9. For some non-empty represented space Y (containing a computable point), the map π2 : A (X × Y) → A (Y) is well-defined and continuous (computable).

Proof. 1. ⇔ 2. IsEmptyX and IsFullX have exactly the same realizers. 1. ⇔ 3. X is always a computable element of A (X), as it is realized by the constant function p 7→ 0N. This provides the implication 1. ⇐ 3.. For the other direction, note IsEmptyA(B) = IsEmptyX(A∩ B), so continuity (computability) of IsEmptyA follows from Proposition 6 (2) together with Corol- lary 4 and Proposition 3 (2).

8For the idea of exhaustive search of infinite sets, see ESCARDO´ [26]. 9We remind the reader that we call A ∈ A (X) computable, if it is a computable name. This generalizes Weihrauch’s notion of a co-c.e. closed set, not that of a computable closed set! 8 Theory of Represented Spaces

1. ⇔ 4. /0 ∈ O(X) is computable by Proposition 3 (6), so if ⊆ is computable, so is A 7→⊆ (A, /0) by Corollary 4. For the other direction, observe A ⊆ B ⇔ A∩BC = /0,so we may use the combination of Proposition 6 (1, 3), Corollary 4 and Proposition 3 (2) to obtain computability of ⊆ here.

2. ⇔ 5. By Proposition 3 (6) we can compute (U)n∈N from U, and clearly IsFull(U) = IsCover((U)n∈N). S On the other hand, by Proposition 6 (4), we can compute n∈N Un from (Un)n∈N, and IsCover((U)n∈N) = S IsFull( n∈N Un). 5. ⇒ 6. Assume that the Turing machine M computes IsCover (potentially with access to some oracle). In order to solve FiniteSubcover, we simulate M on the input for FiniteSubcover (which is of a suitable type). As we know that the input sequence does cover X, we also know that M has to write a 1 eventually. When M writes the first 1, it has only read some finite prefix of the input. In particular, we may assume that M has no information at all about the Un with n > N for some N ∈ N. Moreover, we can find such an N effectively from observing the simulation of M.

Now N constitutes a valid answer to FiniteSubcover((Un)n∈N). To see this, assume the contrary. SN 0 0 0 Then n=0 Un 6= X. Now consider the sequence (Un)n∈N with Un = Un for n ≤ N and Un = /0 0 otherwise. On some name for (Un)n∈N, M will eventually print a 1 - as it cannot distinguish 0 0 (Un)n∈N from (Un)n∈N before that. But as (Un)n∈N does not constitute a cover of X, this contradicts the initial assumption. 6. ⇒ 2. Assume that the Turing machine M computes FiniteSubcover (potentially with access to some

oracle). In order to solve IsFull(U), we simulate M on input (U)n∈N, which we can obtain by Proposition 3 (6). Beside the simulation, we repeatedly write 0s on the output tape. If M ever produces some N ∈ N as output, we write a 1. This actually solves IsFull(U) correctly.

If U = X, then (U)n∈N is a valid input for FiniteSubcover, so eventually some N ∈ N will be produced, causing the final result to be > ∈ S. Now assume that M produces some N ∈ N on input (U)n∈N for U 6= X. At the time where N has been written, M has only received information about some finite prefix of its input. In particular, there is some K > N s.t. M exhibits the same behaviour 0 0 0 when faced with the input sequence (Un)n∈N with Un = U for n ≤ K and Un = X otherwise. As 0 (Un)n∈N is a valid input for FiniteSubcover, M would be required to produce some k ∈ N with k ≥ K rather than N. The contradiction can only be resolved by the assumption that M never

completes an output on input (U)n∈N for U 6= X, which yields the final answer to be ⊥ ∈ S. 6. ⇔ 7. This follows via Proposition 6 (1). 1. ⇒ 8. By Proposition 3 (2), we can compute y 7→ (x 7→ A(x,y)) from A ∈ A (X × Y). Now we find (x 7→ A(x,y)) ∈ A (X), and if IsEmptyX is computable, so is y 7→ IsEmptyX(x 7→ A(x,y)) = π2(A). 8. ⇒ 9. Just instantiate with Y = N. C 9. ⇒ 1. Let y ∈ Y be a (computable) point. Then A 7→ (y ∈ π2(A × Y) ) realizes IsEmptyX, and is continuous (computable) using the assumption, together with Proposition 6 (1, 6, 7) together with Corollary 4.

Proposition 9. Let X be (computably) compact and f : X → Y a (computable) continuous surjection. Then Y is (computably) compact.

Proof. We use the characterization of compactness provided by Proposition 8 (2). By Proposition 6 (6) −1 −1 we find f : O(Y) → O(X) to be continuous (computable). Now observe IsFullY(U) = IsFullX( f (U)) due to surjectivity of f . A. Pauly 9

Proposition 10. If X, Y are (computably) compact, then so is X × Y.

Proof. Note IsEmptyX×Y = IsEmptyY ◦π2, and use Proposition 8 (8). We can call a subset of a represented space compact, if it is compact when represented with the T subspace representation. A set K ⊆ X is called saturated, iff K = {U∈O(X)|K⊆U} U. Noting that for compact K the set {U ∈ O(X) | K ⊆ U} is open in O(X), we obtain a representation of the set K (X) of saturated compact sets by identifying it as a subspace of O(O(X)). In particular, this makes IsContainedIn : K (X) × O(X) → S computable, and provides a uniform counterpart to the results in 8. T For arbitrary sets K ⊆ X, we use ↑K := {U∈O(X)|K⊆U} U to denote its saturation, and point out that any ↑K is saturated. In a T1 space, i.e. a represented space X where ∀x ∈ X {x} ∈ A (X), any set is already saturated. We proceed to exhibit a number of computable operations on spaces of (saturated) compact sets, some of which can be seen as uniform counterparts of results about compact spaces above. Proposition 11. The following operations are well-defined and computable: 1. IsContainedIn : K (X) × O(X) → S 2. x 7→ ↑{x} : X → K (X) 3. ∪ : K (X) × K (X) → K (X) 4. ↑∩ : K (X) × A (X) → K (X) 5. f −1 7→ ↑ f :⊆ C (O(Y),O(X)) → C (K (X),K (Y)); here and below ↑ f is the composition of f and the saturation operator ↑ 6. f 7→ ↑ f : C (X,Y) → C (K (X),K (Y)) 7. ( f ,K) 7→ ↑ f [K] : C (X,Y) × K (X) → K (Y) 8. × : K (X) × K (Y) → K (X × Y)

9. π1 : K (X × Y) → K (X), π2 : K (X × Y) → K (Y)

Proof. 1. Taking into account the definition of K , this is an instantiation of eval from Proposition 3 (1). 2. Note that x ∈ U iff ↑{x} ⊆ U for U ∈ O(X). 3. ∪ : K (X) × K (X) → K (X) is realized by ∩ : O(O(X)) × O(O(X)) → O(O(X)) from Propo- sition 6 (2). 4. The core observation is that A ∩ B ⊆ U iff A ⊆ (U ∪ BC) together with Proposition 6 (1, 2). 5. It suffices to show that f −1 7→ f is well-defined, its computability then follows from the com- putability of f −1 7→ ( f −1)−1 :⊆ C (O(Y),O(X)) → C (O(O(X)),O(O(Y))) as it is a restriction of this map. The computability of the latter map in turn is a special case of Proposition 6 (6). Well- definedness in turn follows directly from the characterization of K (X) as subspace of O(O(X)). 6. From (5) together with Proposition 6 (6). 7. From (6) via type conversion (Proposition 3). 8. We show that given A ∈ K (X), B ∈ K (Y) and U ∈ O(X × Y), we can semidecide A × B ⊆ U. Drawing from (1) and Proposition 6 (9), we consider y 7→ IsContainedIn(A,Cut(y,U)), which defines some element UA of O(Y). Furthermore, we note that A × B ⊆ U iff B ⊆ UA. 10 Theory of Represented Spaces

9. By Proposition 1 the maps π1 : X × Y → X, π2 : X × Y → Y are computable. Corollary 4 allows us to use (6) to lift this to compact sets.

Corollary 12. X is (computably) compact, iff ↑id : A (X) → K (X) is well-defined and continuous (computable).

Remarks

The represented space K (X) was introduced, and various parts of Proposition 11 were proven by Schroder¨ in [54, Section 4.4.3] 2002 (for admissible X). This definition of compactness also follows the approach of synthetic topology (ESCARDO´ 2004 [25]), and some of our results have also been ob- tained there (Proposition 8 [2. ⇒ 3.], Propositions 9, 10). The equivalence in Proposition 8 [2. ⇔ 8.] was shown by ESCARDO´ in [28]. That K (X) can be identified with a certain subspace of O(O(X)) is the statement of the Hofman- Mislove theorem (e.g. [31], see also the extension by SCHRODER¨ in [59]). The role of saturation as canonization operation for compact sets is already discussed by COLLINS [20]. COLLINS has stated the results of Proposition 11 (2. − 4.,6.,7.) as part of [20, Theorem 3.23 & Theorem 3.24]; Corollary 12 as [20, Theorem 3.31 (4)]; and Proposition 8 (1. ⇒ 8.) as [20, Proposition 3.33]. His definition of compactness [20, Definition 3.28 (4)] is equivalent to the one here. However, the claimed characterization of the computably compact spaces as the images of Cantor space com- putable functions in [20, Proposition 3.30] (stated without proof) is wrong. A counterexample (due to DE BRECHT, personal communication) is one-point compactification of NN (10). This also impacts [20, Theorem 3.32]. Some results in this section generalize known results in the far more restricted setting of computable metric spaces. A version of Proposition 8 [5. ⇔ 6.] was proven by BRATTKA and PRESSER (2003 [15]). WEIHRAUCH proved the restricted version of 11 (7) in 2003 [67]. For computable metric spaces, compact compactness can also be characterized by the computability of the outer radius of closed sets (taking values in R>) as shown by LE ROUX and the author as [38, Proposition 19]. The equivalences in Proposition 8 [3. ⇔ 6. ⇔ 7. ⇔ 8.] are uniform counterparts to well-known char- acterizations of compactness in topology. The space K (X) corresponds to the upper Vietoris topology (which is often defined on the hyper- space of closed sets). This topology was introduced by VIETORIS in 1922 [62]. Again, a general source for hyperspace topologies is [5].

6 T2 separation

The T2 separation axiom can, in the context of represented spaces, be understood equivalently as the property of a space making either inequality or the subspace of compact sets well-behaved. The strong connection between T2 separation and compactness is somewhat reminiscent of the ultrafilter approach to topology, where compactness means each ultrafilter converges to at least one point, and T2 that they converge to at most one point. Several of our equivalences will require the T0 property to work, which

10 In general, the one-point compactification of a represented space (X,δ) can be introduced as (X ∪ {⊥},δC) where n δC(0 1p) = δ(p) and δC(0N) = ⊥. Any one-point compactification is computably compact, but can only have a total Cantor space representation if the original space was locally compact. A. Pauly 11 we understand in a non-effective way to mean that x 6= y implies ↑{x} 6= ↑{y}. We also need the (non- effective and non-uniform) T1 property, which we understand to mean that if x 6= y, then ↑{x}∩↑{y} = /0.

Definition 13. A represented space X is (computably) T2, if the map x 7→ {x} : X → A (X) is well-defined and continuous (computable). Proposition 14. — The following properties are equivalent for a represented space X:

1. X is (computably) T2.

2. id : K (X) → A (X) is well-defined and continuous (computable), and X is T0.

3. ∩ : K (X) × K (X) → K (X) is well-defined and continuous (computable), and X is T1. 4. x 7→ ↑{x} : X → A (X) is well-defined, injective and continuous (computable). 5. 6= : X × X → S defined by 6=(x,x) = ⊥ and 6=(x,y) = > otherwise is continuous (computable). 11 6. ∆X = {(x,x) | x ∈ X} ∈ A (X × X) (is computable ). 7. Graph : C (Y,X) → A (Y × X) is well-defined and continuous (computable) for any represented space Y.

Proof. 1. ⇒ 2. The map is identical to K 7→ (x 7→ IsContainedIn((K ∩ {x}), /0)). To see that the map is continuous (computable) under the assumption that x 7→ {x} : X → A (X) is continuous (com- putable), first use Propositions 11 (4) to obtain K ∩{x} ∈ K (X), then Proposition11 (1). Being T0 is an obvious consequence. 1. ∧ 2. ⇒ 3. The continuity (computability) of the map in 3. is a direct consequence of the continuity (computability) of the map in 2. and Proposition 11 (4). The T1 property follows from the well- definedness of x 7→ {x} : X → A (X).

3. ⇒ 2. Observe that x ∈ K holds for a compact set K, if and only if ↑{x} ⊆ K holds. The T1-property allows us to strengthen this to x ∈ K iff ↑{x} ∩ K 6= /0. Hence id : K (X) → A (X) is given by K 7→ (x 7→ IsContainedIn(K ∩↑{x}, /0)), if intersection of compact sets is continuous (computable) (using Propositions 11 (1)). 2. ⇒ 4. Continuity (computability) and well-definedness of the map in 4. follows immediately from con- tinuity (computability) and well-definedness of the map in 2. and Proposition 11 (2). Injectiveness of the map is equivalent to the T0-property. 4. ⇒ 2. Similar to 3. ⇒ 2.: In K 7→ (x 7→ IsContainedIn(K ∩ ↑{x}, /0)) use x 7→ ↑{x} : X → A (X), and the intersection from Proposition 11 (4). 2. ⇒ 5. Given x,y ∈ X, compute ↑{x},↑{y} ∈ K (X) by Proposition 11 (2), and then ↑{x},↑{y} ∈ A (X) by the assumption. Now the claim follows from x 6= y iff x ∈/ ↑{y} ∨ y ∈/ ↑{x}. The equivalence holds due to the T0 property, and the right hand side is computable due to Proposition 5. 5. ⇒ 1. By Proposition 3 (2) we find x 7→ (y 7→=( 6 x,y)) to be continuous (computable), but this has the same realizers as x 7→ {x}. 5. ⇔ 6. This is just a reformulation along the definition of A (X × X). 5. ⇒ 7. This follows from Graph( f ) = {(y,x) | f (y) = x} ∈ A (Y × X); that f × id : Y × X → X × X is available as a continuous function; and Proposition 6 (1,6).

11Again, we remind the reader that we call A ∈ A (X) computable, if it is a computable name. This generalizes Weihrauch’s notion of a co-c.e. closed set, not that of a computable closed set! 12 Theory of Represented Spaces

7. ⇒ 6. Pick Y := X, and then consider Graph(idX).

At the first glance, Definition 13 seems to be a prime candidate for T1 separation rather than T2 separation. In light of Proposition 14, it is clear that the denotation T2 is justified, too. The reason to prefer the identification as T2 rather than T1 separation lies in the observation that a represented space is T2, if and only if the induced topological space is sequentially T2 (a topological space is sequentially T2 iff its diagonal is sequentially closed (e.g. [59, Section 2.4]), so the claim is Proposition 14 (6)). On the 12 other hand, there are represented spaces not satisfying Definition 13 but inducing a T1 topology . With both compactness and the computable T2 property in place, we can proceed to discuss com- putably proper maps, i.e. those (continuous) maps where the preimages of compacts are compact. First, we present a counterpart to a classic result from topology:

Proposition 15. Let X be (computably) compact and Y (computably) T2. Then the continuous maps in C (X,Y) are uniformly proper, i.e. the map ( f ,K) 7→ f −1(K) : C (X,Y) × K (Y) → K (X) is well- defined and continuous (computable).

Proof. By Proposition 14 (2) the map id : K (Y) → A (Y) is continuous (computable) due to the T2- property for Y. The map ( f ,A) 7→ f −1(A) : C (X,Y) × A (Y) → A (X) is computable by Proposition 6 (1, 6), Proposition 3 (1). By Corollary 12 the map id : A (X) → K (X) is continuous (computable) due to the compactness of X. Composition of these maps yields the claim.

Proposition 16. Let f : X → Y be surjective, continuous (computable) and (computably) proper, i.e. let −1 f : K (Y) → K (X) be well-defined and continuous (computable). If X is (computably) T2 and Y is T1, then Y even is (computably) T2.

Proof. We use the characterization of the T2 property given in Proposition 14 (3), i.e. the intersection of compact sets being computable as a compact set. For surjective f , we have A ∩ B = f [ f −1(A) ∩ f −1(B)]. Using this equation for A,B ∈ K (Y), we can compute f −1(A), f −1(B) by the assumption f −1 −1 were computably proper, then f (A) ∩ f (B) as a compact set by assumption X were T2, and finally ↑ f [ f −1(A) ∩ f −1(B)] ∈ K (Y) due to Proposition 11 (6). Thus, we see that we can obtain ↑(A ∩ B) ∈ K (Y) from A,B ∈ K (Y). By inspecting the proof of Proposition 14 (3. ⇒ 2.), we notice that we only care whether the intersection of two compact sets is empty or not. As saturation preserves the empty set, we conclude that computability of ↑∩ : K (Y) × K (Y) → K (Y) suffices for a T1 space Y to be computably Hausdorff.

Corollary 17. If X is T1 and has a computably proper representation δX, then X is computably T2.

Remarks

ESCARDO´ uses the condition Proposition 14 (5) to introduce the Hausdorff property in [25]; and proceeds to prove Proposition 14 [5. ⇒ 2.].WEIHRAUCH studied computable separation in countably based spaces in [69] (2010), including what corresponds to the equivalence Proposition 14 [1. ⇒ 5.]. COLLINS translated ESCARDO´ ’s definition of Hausdorff and the equivalence from Proposition 14 [5. ⇒ 2.] into computable analysis in [20, Definition 3.28 (2 & Theorem 3.31 (2.)] (under the name effectively distinguishable).

12 An example is the subspace {{n} | n ∈ N} ⊆ A (N), as pointed out by WEIHRAUCH in [69]. This space corresponds to the cofinite topology on N. A. Pauly 13

Proper representations were studied by SCHRODER¨ in [57] 2004; in particular, [57, Theorem 5.3] shows that having a proper admissible representation even implies metrizability (and is implied by metrizability, in turn); which is much stronger than Corollary 17. A minor further strengthening of the result can be found in [36], where it is shown that any such space embeds computability into a computable metric space.

7 Overtness

Like compactness, overtness both is a property of represented spaces and induces a space of certain subsets. Unlike compactness, overtness is classically vacuous and thus not that well-known. As discussed below, overtness has been present in the study of representations of spaces of subsets in computable analysis for a long time, although in a disguised role.ˆ

Definition 18. Let IsNonEmptyX : O(X) → S be defined by IsNonEmptyX(/0) = ⊥ and IsNonEmptyX(U) = > for U 6= /0.Now call X (computably) overt, iff IsNonEmptyX is continuous (computable). This definition shows that overtness is in some sense the dual to compactness. In computable anal- ysis, typically the criterion separable rather than overt has been used for spaces, based on the following observation:

Proposition 19. Let (an)n∈N be a dense sequence in X. Then IsNonEmptyX is computable relative to (an)n∈N.

Proof. If U ∈ O(X) is non-empty, it contains some ak. Thus simultaneously testing ai ∈ U for all i ∈ N will detect non-emptyness of U, if true.

As all represented spaces admit a dense sequence (by lifting a dense sequence in the domain of the representation), we see that all represented spaces are overt – merely computable overtness survives as a distinguishing criterion.

Proposition 20. Let X be (computably) overt and Y be (computably) T2. Then C (X,Y) is (computably) T2.

Proof. For f ,g : X → Y we find f 6= g ⇔ IsNonEmptyX({x ∈ X | f (x) 6= g(x)}).

Similar to our procedure for compact sets, we can introduce the space V (X) of overt sets. The crucial mapping will be Intersects : V (X)×O(X) → S defined by Intersects(A,U) = 1 iff A∩U 6= /0.This makes clear that V (X) can be understood as a subspace of O(O(X)). A set O ∈ O(O(X)) will correspond to an element of V (X), iff it is of the form OA = {U ∈ O(X) | U ∩ A 6= /0} for some subset A ⊆ X. Note that OA = OB for two sets A,B ⊆ X, iff A = B (here denotes the topological closure). Hence, we obtain V (X) by interpreting each set of the form OA ∈ O(O(X)) as A ⊆ X. In particular, we see that the overt sets and the closed sets coincide extensionally. In fact, the equiv- alence class of representations for V (X) has been studied for a long time, called the representation of closed sets by positive information. It is known that this representation is incomparable to the represen- tation by negative information, i.e. that neither id : V (X) → A (X) nor id : A (X) → V (X) is continuous for non-empty X. Our justification to avoid the word closed for the space V (X) lies in the computable structure available on it, which differs significantly from the usual closure properties expected from closed sets: 14 Theory of Represented Spaces

Proposition 21. Let X, Y, Z be represented spaces, and let Z be computably overt. Then the following functions are computable: 1. : O(Z) → V (Z) 2. x 7→ {x} : X → V (X) 3. ∪ : V (X) × V (X) → V (X) S 4. : C (N,V (X)) → V (X) 5. ∩ : V (X) × O(X) → V (X)

6. π1 : V (X × Y) → V (X) 7. ( f ,A) 7→ f [A] : C (X,Y) × V (X) → V (Y) 8. ( f −1,A) 7→ f [A] :⊆ C (O(Y),O(X)) × V (X) → V (Y), which takes a function between open sets that is the preimage map obtained from some (topologically continuous) function f : X → Y and an overt subset A of X, and outputs the closure of the image of A under f as an overt subset of Y

Proof. 1. Intersects(V,U) = IsNonEmptyZ(U ∩V), using Proposition 6 (3) for ∩ : O(Z) × O(Z) → O(Z) 2. Intersects({x},U) = (x ∈ U), using Proposition 6 (7) for ∈ : X × O(X) → S 3. Intersects(A ∪ B,U) = ∨(Intersects(A,U),Intersects(B,U)) S W 4. Intersects( n∈N An,U) = n∈N(Intersects(An,U)) 5. Intersects(V ∩ A,U) = Intersects(A,V ∩U), using Proposition 6 (3) for ∩ : O(X)×O(X) → O(X)

6. Intersects(π1(A),U) = Intersects(A,U × Y), using Proposition 6 (8) for × : O(X) × O(Y) → O(X × Y) 7. Intersects( f [A],U) = Intersects(A, f −1(U)), using Proposition 6 (6) to obtain f −1 : O(Y) → O(X) 8. Intersects( f [A],U) = Intersects(A, f −1(U))

Remarks

The notion of overtness was named as such by TAYLOR, the definition here follows ESCARDO´ [25]. This reference also contains Proposition 20 and Proposition 21 (7). The statements of Proposition 19 and 21 (5) are very similar to [4, Proposition 2.2] by BAUER and LESNIK (2012). In restricted settings such as Euclidean spaces, WEIHRAUCH has studied V (X) as the closed subsets of X represented by positive information, and e.g. proven Proposition 21 (3) in [66]. The representation of V (X) via an identification with a subspace of O(O(X)) was done by SCHRODER¨ in [54, Section 4.4.2] (2002), together with several of its closure properties. Both ESCARDO´ and TAYLOR have pointed out that classically, every space is overt. COLLINS translated the definition of overtness into the language of computable analysis (under the name effectively separable, and introduced the notation V (X) in [20] (2010). He discusses the role of topological closure as the canonization operation for the overt sets; and provides the results of Proposition 21 (1.−5.,7.). His claimed characterization of the computably overt spaces as those having a computable dense sequence in [20, Proposition 3.30] (stated without proof) is wrong, though. A counterexample is A. Pauly 15 the space {p ∈ {0,1}N | p is not computable} understood as a subspace of {0,1}N. The failure of [20, Proposition 3.30] then impacts [20, Theorem 3.32]. The space V (X) corresponds to the lower Vietoris topology (or, equivalently, the lower Fell topol- ogy) (as shown by SCHRODER¨ ). This topology was introduced by VIETORIS in 1922 [62]. Again, a general source for hyperspace topologies is [5]. Thus, we can obtain the Vietoris topology on a space of subsets as K (X) ∧ V (X) and the Fell topology as A (X) ∧ V (X). Both of these spaces appear in a variety of results.

8 Discreteness

Just as the T2 separation property could be characterizes as making intersection of compact sets com- putable, there is a property of a space that makes intersection of overt sets computable. This property, discreteness, is in many ways the dual to T2 separation, just as overt is the dual to compact. Definition 22. A represented space X is called (computably) discrete, iff the map x 7→ {x} : X → O(X) is well-defined and continuous (computable). Theorem 23. The following properties are equivalent for a represented space X: 1. X is (computably) discrete.

2. id : V (X) → O(X) is well-defined and continuous (computable), and X is T1.

3. ∩ : V (X) × V (X) → V (X) is well-defined and continuous (computable), and X is T1. 4. = : X × X → S defined by =(x,x) = > and =(x,y) = ⊥ otherwise is continuous (computable).

5. ∆X = {(x,x) | x ∈ X} ∈ O(X × X) (is computable).

Proof. 1. ⇒ 2. We find x ∈ A iff Intersects(A,{x}). The (topological) T1 property is a straightforward consequence of singletons being open. 2. ⇒ 3. Use ∩ from Proposition 21 (5) together with the assumption. 3. ⇒ 4. Given (x,y) ∈ X × X, we can use Proposition 21 (2) to compute ({x},{y}) ∈ V (X) × V (X). The T1 property means {x} = {x} and {y} = {y}. Subsequently we use ∩ from the assumption to compute {x} ∩ {y} ∈ V (X), and then Intersects({x} ∩ {y},X), which is identical to =(x,y). 4. ⇒ 1. This is a consequence of currying to x 7→ (y 7→ =(x,y)). ∼ 4. ⇔ 5. This is straightforward using C (X × X,S) = O(X × X).

As any represented space is separable (as pointed out in Section 7), the requirement that {x} ∈ O(X) for any x ∈ X already forces X to be countable, thus producing the following: Proposition 24. Any discrete represented space is countable. It is worth pointed out that – unlike in topology – being computably discrete does not imply being computably Hausdorff. A counterexample can be constructed by starting with a recursively enumerable but not recursive set A ⊆ N, and then identifying the elements of A. This means we define a representation δ : NN → (N \ A) ∪ {A} by δ(p) = A if p(0) ∈ A and δ(p) = p(0) if p(0) ∈/ A. This makes equality recognizable, but not refutable. Proposition 25. Let X be (computably) compact and Y be (computably) discrete. Then C (X,Y) is (computably) discrete. 16 Theory of Represented Spaces

Proof. We find =( f ,g) = IsFullX({x ∈ X | =( f (x),g(x))}), the claim now follows with Theorem 23 (4) and Proposition 8 (2).

Unlike for its dual result in Proposition 20, it is easy to find a counter-example showing the necessity of a restriction on the domain in the preceding result. As such, consider X = Y = N. The space N is ∼ computably discrete, but not compact, and C (N,N) = NN is not discrete by Proposition 24.

Remarks

ESCARDO´ uses the characterization in Theorem 23 (4) as definition of discreteness, and proceeds to prove Theorem 23 [4. ⇒ 2.] and Proposition 25 in [25]. That computable discreteness does not imply computably Hausdorff has been observed by both ESCARDO´ ; and by WEIHRAUCH in [69]. In the setting of computable analysis, COLLINS introduced discreteness and proved the equivalence from Theorem 23 (1. ⇔ 2.) as [20, Theorem 3.31 (1)].

9 Admissibility as effective T0 separation

Recall the computable maps x 7→ ↑{x} : X → K (X) (Proposition 11 (2)) and x 7→ {x} : X → V (X) (Proposition 21 (2)), and note that both are realized in the same way: An element x ∈ X provides the abstract capacity to determine membership in an open set (since x ∈ U ⇔ ↑{x} ⊆ U ⇔ {x} ∩U 6= /0).

Definition 26. Consider the computable map κX : X → O(O(X)) defined by κX(x) = {U ∈ O(X) | x ∈ U}. Let Xκ denote the image of X in O(O(X)) under κX.

With the considerations above, we see that we may consider Xκ simultaneously as a subspace of K (X) and V (X). If we understand Xκ as a subspace of K (X), then κX(x) = ↑{x}, if as a subspace 13 of V (X), then κX(x) = {x}. This identification will be helpful to see that κ is an endofunctor on the category-extension of represented spaces. That κ commutes with composition is rather obvious, hence we only need the following:

Proposition 27. There is an induced computable map κ : C (X,Y) → C (Xκ ,Yκ ) for all represented spaces X, Y such that the following diagram commutes:

f X −−−−→ Y   κ κ y X y Y κ( f ) Xκ −−−−→ Yκ

Proof. This follows e.g. from Proposition 11 (6) and Xκ ⊆ K (X), together with ↑ f [↑{x}] = ↑{ f (x)}.

In general, κX : X → Xκ will fail to be (computably) continuously invertible, however, those spaces admitting a continuous (computable) left-inverse for κX can be characterized as exactly those fully un- derstandable in terms of their topology.

Definition 28. Call X (computably) admissible, iff κX : X → Xκ admits a continuous (computable) left- inverse.

13In fact, we could even call κ a computable endofunctor, as its restriction to any homset is computable. A. Pauly 17

We will proceed to demonstrate that κ is the coreflector of the (cartesian-closed) subcategory of the (computably) admissible represented spaces inside the category of represented spaces. In a way, this coreflector is analogous to the Kolmogorov-quotient (or T0-coreflector) in topology. Corollary 29. Let X, Y be represented spaces, and let Y be (computably) admissible. There is a (com- putable) continuous map R : C (X,Y) → C (Xκ ,Y) such that f = R( f ) ◦ κX for all f ∈ C (X,Y).

Proof. Take the map κ : C (X,Y) → C (Xκ ,Yκ ) from Proposition 27. Then we use Proposition 3 (4, 6) −1 to compose κ( f ) with κY obtained from Definition 28.

Proposition 30. S is computably admissible.

Proof. We can explicitly define κ−1 :⊆ ( ( )) → by κ−1(U) = ({>} ∈ U). S O O S S S

Theorem 31. Let Y be (computably) admissible. Then C (X,Y) is (computably) admissible.

Proof. Given some U ∈ O(Y) and x ∈ X, we can compute { f ∈ C (X,Y) | f (x) ∈ U} ∈ O(C (X,Y)). Hence, given x ∈ X and {U ∈ O(C (X,Y)) | f ∈ U} ∈ O(O(C (X,Y))) we can compute {U ∈ O(Y) | f (x) ∈ U} ∈ O(O(Y)) by Proposition 6 (6). By assumption, the latter suffices to compute f (x) ∈ Y. Currying yields the claim.

Corollary 32. Xκ , O(X), A (X), K (X) and V (X) are all computably admissible. ∼ ∼ ∼ ∼ Corollary 33. O(X) = O(Xκ ), A (X) = A (Xκ ), K (X) = K (Xκ ) and V (X) = V (Xκ ).

Corollary 34. κX is effectively open, i.e. there is a well-defined and computable map K : O(X) → O(Xκ ) such that K(U) = κX[U].

Corollary 35. X is (computably) compact, (computably) T2, (computably) overt or (computably) discrete if and only if Xκ has that property. Theorem 36. The following properties are equivalent for a represented space X: 1. X is (computably) admissible.

2. f 7→ fK : C (Y,X) → C (K (Y),K (X)) has a well-defined and continuous (computable) partial inverse for any represented space Y; where fK(A) =↑ f [A]. 3. f 7→ f −1 : C (Y,X) → C (O(X),O(Y)) has a well-defined and continuous (computable) partial inverse for any represented space Y. 4. Any topologically continuous function f : Y → X (i.e. f −1 : O(X) → O(Y) is well-defined) is con- tinuous as a function between represented spaces (i.e. f ∈ C (Y,X)). (no computable counterpart)

Proof. 1. ⇒ 2. For precision, let fK be the input, i.e. the map from compact sets to compact sets derived −1 from the desired output f . Now note f = κX ◦ fK ◦ κY, and consider Proposition 27, Definition 28 and Proposition 3 (4). 2. ⇒ 3. Given a continuous function of the form f −1 : O(X) → O(Y) induced by some continuous f : Y → X, we may use Proposition 11 (4) to obtain the induced function fK : K (Y) → K (X). By assumption, we can recover f ∈ C (X,Y) from the latter. 18 Theory of Represented Spaces

−1 3. ⇒ 1. First we shall show that κX : Xκ → X is well-defined, using proof-by-contradiction. So let us assume κX were not injective, i.e. there were x 6= y ∈ X with κX(x) = κX(y), hence x ∈ U ⇔ y ∈ U for any U ∈ O(X). Then the constant function x,y : {0} → X with x(0) = x and y(0) = y are continuous and distinct, but still x−1 = y−1 as maps of the type O(X) → O({0}). This contradicts the well-definedness of f −1 7→ f :⊆ C (O(X),O({0})) → C ({0},X).

Consider Y := Xκ . By Corollary 34, κX is effectively open, hence κX ∈ C (O(X),O(Xκ )) is a −1 −1 −1 computable element. As κX is well-defined, κX = (κX ) , and we can use the assumption to −1 obtain κX : Xκ → X from κX as open map. 1. ⇒ 4. By Corollary 33, a function f : Y → X is topologically continuous, iff it is so as a function f : Y → Xκ . We may curry topologically continuous functions, too, and obtain χ f : Y×O(X) → S as a topologically continuous function defined by χ f (y,U) = > iff f (y) ∈U. By composition, so is χ f ◦ N (δY × δO(X)) :⊆ {0,1} → S. But this just means χ f ◦ (δY × δO(X)) ∈ O(dom(δY × δO(X))). Now the process can be reversed, using represented space continuity in place of topological continuity ∼ to find that f ∈ C (Y,Xκ ). Admissibility of X, i.e. X = Xκ , then allows us the make the final step and reach f ∈ C (Y,X). −1 4. ⇒ 1. Again, we first show that κX is well-defined using proof-by-contradiction. Assume there were x 6= y ∈ X with κX(x) = κX(y), hence x ∈ U ⇔ y ∈ U for any U ∈ O(X). Then any function f : {0,1}N → {x,y} ⊆ X is topologically continuous. However, there are 22ℵ0 such functions, whereas |C ({0,1}N,X)| ≤ 2ℵ0 . −1 −1 −1 Using again κX = (κX ) , as well as Corollary 34, we see that κX : Xκ → X is topologically continuous, hence we obtain it as a continuous function between represented spaces.

Note that only the equivalence of 1.,2.,3. in the preceding theorem has a proper place in a synthetic treatment – the reference in 4. to the mere (external) well-definedness of f −1 is meaningless from a strictly internal view on a given category. Its provability here is due to the definition of continuity for represented spaces via (topological) continuity on Cantor space. What we obtain from it is the observa- tion that the admissibly represented spaces (with continuous maps) simultaneously form a subcategory of the represented spaces and of the topological spaces, moreover, that this is in some sense the largest such joint subcategory. As an example for the interplay of admissibility and some of the other properties of represented spaces, we shall briefly revisit the connection between a function and its graph (cf. Proposition 14 (7)): Proposition 37. Let Y be (computably) admissible and (computably) compact. Then Graph−1 :⊆ A (X×Y) → C (X,Y) is continuous (computable), where dom(Graph−1) = {A ∈ A (X×Y) | ∃ f : X → Y Graph( f ) = A}.

Proof. We may assume that Graph( f ) ∈ A (X×Y) and x ∈ X are given and show how to obtain f (x) ∈ Y. Using Proposition 6 (1,9), we obtain {y | (x,y) ∈ Graph( f )} = { f (x)} ∈ A (Y). Corollary 12 gets us ↑{ f (x)} ∈ K (Y), by definition of Yκ thus f (x) ∈ Yκ . By definition of admissibility, this in turn suffices to obtain f (x) ∈ Y.

Similar results can be obtained for compact and overt graphs: −1 Proposition 38. Let Y be (computably) admissible and X (computably) T2. Then Graph :⊆ K (X × Y) → C (X,Y) is continuous (computable), where dom(Graph−1) = {A ∈ K (X×Y) | ∃ f : X → Y Graph( f ) = A}. A. Pauly 19

Proof. We may assume that Graph( f ) ∈ K (X × Y) and x ∈ X are given and show how to obtain f (x) ∈ Y. By Definition 13 we can obtain {x} ∈ A (X), and then {x} ×Y ∈ A (X × Y) by Proposition 6 (8). Then Proposition 11 (4) provides us with ↑(({x} ×Y) ∩ Graph( f )) = ↑({x} × { f (x)}) ∈ K (X × Y). Then we use projection (Proposition 11 (9)) to get ↑{ f (x)} ∈ K (Y), admissibility of Y enables us to extract f (x).

Proposition 39. Let Y be (computably) admissible and X (computably) discrete. Then Graph−1 :⊆ V (X × Y) → C (X,Y) is continuous (computable), where dom(Graph−1) = {A ∈ V (X × Y) | ∃ f : X → Y Graph( f ) = A}.

Proof. We may assume that Graph( f ) ∈ V (X×Y) and x ∈ X are given and show how to obtain f (x) ∈ Y. By Definition 22 we can obtain {x} ∈ O(X), and then {x} ×Y ∈ O(X × Y) by Proposition 6 (8). Then Proposition 21 (5) provides us with ({x} ×Y) ∩ Graph( f ) = {x} × { f (x)} ∈ V (X × Y). Then we use projection (Proposition 21 (6)) to get { f (x)} ∈ V (Y), admissibility of Y enables us to extract f (x).

How to use admissibility Admissibility is at the core of a very common scheme to prove computability of some specific mapping:

1. Conclude that the solution set S is closed based on its definition (this often involves the Hausdorff condition). 2. Obtain some compact candidate set K (either from the situation, or by assumption - this often involves some bounds). 3. Compute ↑(S ∩ K) as a compact set via Proposition 11 (4). 4. Use domain-specific reasoning or assumption to conclude that the solution s is unique. 5. As we have ↑{s} available as a compact set, if our target space is admissible, we can compute s.

While this scheme is usually well-hidden and obscured, it is present e.g. in [30] by GALATOLO, HOYRUP and ROJAS, [52] by RETTINGER, [22, 23] by COLLINS and GRAC¸ A. A very similar algorithm (albeit in a slightly different formal setting) is described by ESCARDO´ in [27]. This use of admissibility also underlies the model of non-deterministic type-2 machines suggest by ZIEGLER [74] and studied further by BRATTKA, DE BRECHT and the author in [11, 14].

Remarks

The notion of admissibility was introduced by KREITZ and WEIHRAUCH in 1985 [37]. They essentially define a represented space to admissible if it is isomorphic to a subspace of O(N), which captures the countably based admissible spaces. Then they proceed to prove that this implies the condition in Theorem 36 (4). The understanding of admissibility as presented here is essentially due to SCHRODER¨ . In [56] (2002) SCHRODER¨ uses the condition in Theorem 36 (4) as definition of admissibility, characterizes the topolo- gies arising as O(X) and in particular proves a slightly weaker version of Theorem 31 (as it requires admissibility of X, too). In his thesis [54] (2002), SCHRODER¨ also defines computable admissibility (i.e. Definition 28) and provides the statements of Propositions 27, 30, Corollary 29 and Theorems 31, 36. LIETZ also added to the development, e.g. [40, Theorem 3.2.7.], which shows that Definition 28 is indeed well-suited for its purpose. 20 Theory of Represented Spaces

The identification of a point x ∈ X with its neighbourhood filter {U ∈ O(X) | x ∈ U} ∈ O(O(X)) is reminiscent of the ultrafilter approach to topology. The study of admissibility can be seen as an attempt to generalize the coincide of computability relative to an oracle and (topological) continuity beyond the setting of Baire space. While dealing only with a restricted class of spaces, [13] by BRATTKA and HERTLING (1994) and [51] by the author and ZIEGLER (2013) consider the questions for multivalued functions rather than just functions. The connections between a continuous function and its graph as exemplified in Propositions 37, 38, 39 has been advanced by BRATTKA in [10] (2008).

10 Open relations and compact or overt sets

The dual nature of the spaces of overt and of compact sets becomes clearer when the interaction with open relations is studied. This in particular generalizes the well-known results about upper and lower computability of the maximum of compact subsets of the real numbers presented either as compact sets, or as overt sets (i.e. by positive information). Proposition 40. The map ∃ : O(X×Y)×V (X) → O(Y) defined by ∃(R,A) = {y ∈Y | ∃x ∈ A (x,y) ∈ R} is computable. Moreover, whenever ∃ : O(X × Y) × S (X) → O(Y) is computable for some hyperspace S (X) and some space Y containing a computable element y0, then : S (X) → V (X) is computable. Proof. To see that ∃ is computable, note that y ∈ ∃(R,A) iff Intersects(A,Cut(y,R)) and use Proposition 6 (9) and the definition of V (X). Now if ∃ is computable for some other hyperspace S (X) in place of V (X), we can compute : S (X) → V (X) via Intersects(A,U) = ∈(y0,∃(U ×Y,A)). S S Corollary 41. : V (O(Y)) → O(Y) is computable. (This functions maps A ∈ V (O(Y)) to ( U∈A U) ∈ O(Y).)

Proof. Pick X = O(Y), and instantiate R with {(U,x) ∈ O(Y) × Y | x ∈ U}.

Proposition 42. The map ∀ : O(X×Y)×K (X) → O(Y) defined by ∀(R,A) = {y ∈Y | ∀x ∈ A (x,y) ∈ R} is computable. Moreover, whenever ∀ : O(X × Y) × S (X) → O(Y) is computable for some hyperspace S (X) and some space Y containing a computable element y0, then ↑id : S (X) → K (X) is computable. Proof. To see that ∀ is computable, note that y ∈ ∀(R,A) iff IsContainedIn(A,Cut(y,R)) and use Propo- sition 6 (9) and Proposition 11 (1). Now if ∀ is computable for some other hyperspace S (X) in place of K (X), we can compute ↑id : S (X) → K (X) via IsContainedIn(A,U) = ∈(y0,∀(U ×Y,A)). T T Corollary 43. : K (O(Y)) → O(Y) is computable. (This function maps A ∈ K (O(Y)) to ( U∈A U) ∈ O(Y).)

Proof. Pick X = O(Y), and instantiate R with {(U,x) ∈ O(Y) × Y | x ∈ U}. To make the connection to the computability of maxima on the reals, we first generalize the Dedekind- cut construction of the reals. Let ≺ ∈ O(X × X) be some transitive and open relation. Now we define X≺ := {U ∈ O(X) | ∀x ∈ X(∃y ∈ U x ≺ y ⇒ x ∈ U)} as the subspace of O(X) containing the open initial segments of ≺. Next, consider the computable map x 7→ Cut(x,≺) : X → X≺ mapping x to {y ∈ X | y ≺ x}. We shall denote its image by X≺, and identify an element x with its ≺-lower ideal. Depending on the properties of ≺, the map id : X → X≺ induced by such an identification may fail to be injective. Spaces of this form that have been studied so far are R< and R>, the former in particular playing a central roleˆ A. Pauly 21

in computable measure theory (e.g. SCHRODER¨ [58], COLLINS [21]). By construction, the spaces X≺ and X≺ are computably admissible.

Proposition 44. 1. ≺∈ O(X × X≺), ≺∈ O(X × X) are computable.

2. (x,y) 7→ {z ∈ X | x ≺ z ≺ y} : X × X≺ → O(X) is computable. 14 3. If ≺ is dense and X is (computably) overt, then ≺∈ O(X × X≺) (is computable).

4. sup≺ : V (X) → X≺ and sup≺ : K (X) → X are computable. 5. Let Y ⊆ X be a dense15 and computably overt subspace, and ≺ be dense. Then we can identify Y≺ and X≺.

Proof. 1. ≺: X×X≺ → S is a restriction of the computable map ∈: X×O(X) → S from Proposition 6 (7). The second claim follows by symmetry. 2. This map is a restriction of the computable map ∩ : O(X)×O(X) → OX) from Proposition 6 (3).

3. If ≺ is dense, then x ≺ y iff IsNonEmptyX({z ∈ X | x ≺ z ≺ y}) which is computable by 2. and Definition 18. 4. These are corollaries of Propositions 40, 42.

5. The restriction of any element of X≺ to Y will yield an element of Y≺. Under the conditions given, we can recover a lower ideal U ∈ X≺ from its restriction to Y by noting x ∈ U ⇔ ∃y ∈ Y x ≺ y∧y ∈ U.

If ≺ is e.g. a linear order, then the identification id : X → X≺ is injective, however, will usually not be computably invertible. The best we can obtain here is the following: Proposition 45. Let the intervals of ≺ be an effective base for O(X), i.e. let the computable map S (ai,bi)i∈N 7→ i∈ {x | ai ≺ x ≺ bi} : C (N,X × X) → O(X) be surjective and admit a computable right- N ∼ inverse (as a multivalued function). Then Xκ = (X≺ ∧ X ).

Proof. As X≺ and X are admissible, the identification maps id : Xκ → X≺ and id : Xκ → X are well- defined and computable. It only remains to be shown that they admit a joint computable right-inverse. For this, we need to demonstrate how given x ∈ X≺ ∧ X and U ∈ O(X) we can recognize x ∈ U?. By S assumption, U can be effectively expressed as i∈N{y | ai ≺ y ≺ bi}, and by Proposition 44 (1) we can simultaneously semidecide ai ≺ x and x ≺ bi, until for some i ∈ N both statements are true, in which case x ∈ U is accepted.

We just briefly introduce the represented space R to provide an actual example. In this, let νQ be a N standard notation of the rationals. Then we define ρR :⊆ N → R by ρR(p) = x iff ∀n ∈ N . d(νQ(p(n)),x) < −n 2 ; and consider ρR as the representation of R. We find <∈ O(R×R) to be computable, thus R is com- putably T2. Moreover, R satisfies the conditions of Proposition 45. ∼ ∼ ∼ Corollary 46. R≺ = Q≺, R = Q , R = R≺ ∧ R . Corollary 47. max : K (R) ∧ V (R) → R is computable. Corollary 48. max-value : (K (R) ∧ V (R)) × C (R,R) → R is computable, where max-value(A, f ) = max{ f (x) | x ∈ A}.

14Here, dense is used in the order-theoretic sense, i.e. ≺ is dense means that ∀x,y ∈ Xx ≺ y ⇒ (∃z ∈ X x ≺ z ≺ y). 15And here dense is used in the topological sense, i.e. means∀U ∈ O(X)U 6= /0 → ∃x ∈ Y ∩U. 22 Theory of Represented Spaces

Proof. By Proposition 11 (7) and Proposition 21 (7), we can compute f [A] ∈ (K (R) ∧ V (R)), then we use max from Corollary 47.

Remarks

Corollaries 41 and 43 are present in ESCARDO´ ’s [25] (2004). The topological counterpart of Corol- lary 43 has been described by NACHBIN [43] in 1992. The results on R< and R> are e.g. included in WEIHRAUCH’s [66] (2000).

11 Concluding remarks

It was demonstrated that compactness (Proposition 8), T2 separation (Proposition 14), overtness and discreteness (to a lesser extent), and admissibility (Theorem 36) all encompass various equivalent prop- erties of represented spaces each characterized by the continuity or computability of specific maps. This marks the extent of possible generalizations of many standard results on computability for sets and func- tions. In particular, unnecessary restrictions in prior work such as admissibility, second-countability or metrizability are avoided. As the realizers witnessing the computability of the relevant mappings are generic and independent of the represented spaces involved, the approach presented here is a viable alternative to the introduction of multi-representations of countably based admissibly represented spaces suggested in [53] by RET- TINGER and WEIHRAUCH. Moving away from effective topological spaces, i.e. countably based admissible represented spaces, to represented spaces as the primitive objects of investigation for set and function computability opens up options for an integration with the study of hyper-computation. ZIEGLER [75, 76] and BRATTKA (see [11]) observed that various kinds of hyper-computation such as limit computability or computability with mindchanges can be adequately characterized by means of operators generating new represented spaces from given ones. Based on these, a suitable characterization of functions such as topological closure and interior in terms of computable maps between appropriate represented spaces ought to be possible along the lines of [12]. A more general approach to such jump operators can be found in [17] by DE BRECHT. These operators (which amount to endofunctors) form the basis of a new approach to descriptive set theory [50, 49] as suggested by PAULY and DE BRECHT, which in particular provides derived represented 0 spaces e.g. for the Σn-measurable functions between given represented spaces, thus generalizing [9] by BRATTKA. In this, the applicability of descriptive set theory is extended even further than to the Quasi- Polish spaces introduced by DE BRECHT in [18]. The abstract study of represented spaces seems to hold further aspects to contemplate. Proposition 45 already made use of the concept of an effective basis, and a systematic study of those seems to be promis- ing also in order to obtain results on product spaces, e.g. a version of the Tychonoff theorem. Computable measure theory also is amenable to a similar development as computable topology underwent here, as demonstrated by COLLINS in [21].

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Acknowledgements

I would like to thank Vasco Brattka, Martin Escardo´ and Matthias Schroder¨ for various discussions crucial for the formation of the present work. The article benefitted from comments by Eike Neumann, Greg Yang and Klaus Weihrauch, as well as the anonymous referees.